DEPARTMENT OF CIVIL ENGINEERING

M.E STRUCTURAL ENGINEERING

ST7201 – FINITE ELEMENT ANALYSIS

QUESTION BANK

YEAR/SEM: I/II

PART-A UNIT-I 1. What is the basic concept of finite element analysis? 2. What is meant by ‘discretization’? 3. What is the requirement of displacement field to be satisfied in the use of Rayleigh –Ritz method? 4. Give examples of Eigen value problems in structural mechanics. 5. How will you classify essential and non-essential boundary condition? 6. Derive thermal load vector for 1D bar element. 7. Explain body force and surface force with examples. 8. What do you mean by convergence in finite element analysis? 9. What is the significance of weak formulation? 10. What is quadratic shape function? 11.Why polynomial shape functions are preferred? 12. What are the limitations of Galerkin formulation? 13. Write down the stiffness matrix for 2D beam element. 14. What is shape function? 15. What is displacement and shape function? 16. What is Ritz technique? 17. Write the potential energy for beam of span ‘L’ simply supported at ends, subjected to a concentrated ‘P’ at midspan. Assume EI constant. 18. What are the advantages of FEA? 19. Define banded structure? 20. Define the modulus of resilience?

PART-B

1. Solve the following equations by Gauss elimination method?

i) 28r1+6r2 =1

ii) 6r1+ 24r2+6r3=0

iii) 6r2+28r3+8r4= -1

iv) 8r3+16r4 = 10

2. A simply supported beam is subjected to uniformly distributed load over entire span.

Determine the bending moment and deflection at the mid span using Rayleigh –Ritz method

and compare with exact solution. Use a two term trial function y=a1sin(πx/l)+a2sin(3πx/l).

3. Discuss Rayleigh –Ritz and Galerkin methods of formation by taking an example.

4. Show from first principle that the stiffness matrix of a general finite element can be

evaluated in the form of K=ʃ BTCBdr

V

5. Derive stiffness matrix for a one dimensional bar subjected to both distributed load and point loads. (Note: Differential equation should be formed and then apply basic Galerkin

method on differential equation to form element stiffness matrix.)

6. Solve the following differential equation using Ritz method.

d2y/dx

2 = -sin (πx) boundary conditions u(0) = 0 and u(1) = 0.

7.i) Taking a differential equation , explain the process of weak formulation. ii) Explain any two methods of weighted residuals with examples.

8. i)Derive the element stiffness matrix and element force matrices for a one dimensional line

element. ii) A simply supported beam of span L, young’s modulus, moment of inertia I is subjected to

a uniformly distributed load of P/unit length. Determine the deflection W at the midspan. Use

Rayleigh Ritz method.

9.For the bar shown in fig, determine the nodal displacement, stress in each material and

reaction forces.

P

1. Aluminum 2.Steel

A1= 2400 mm2 A2= 600mm

2 P= 200kN

E1= 70GPa E2= 200GPa

10. Derive shape functions for a 2D beam element.

11. Analyze the frame shown in fig. EI= constant.

B 10kN/m C

3m

3m

A

UNIT – II

1. Define the concept of potential energy?

2. Define ‘Natural coordinate system’?

3. What are the properties of shape functions?

4. What are the advantages of expressing displacement field in Natural co-ordinates than

generalized co-ordinates?

5. Specify stress and strain tensors for plane stress case. Give suitable examples for plane

stress problems.

6. Derive transformation equation used in Gaussian integration

7. What is natural or intrinsic coordinate?

8. What do you mean by isoparametric formulation?

9. Explain plane strain problem with an example.

10. Give any two examples where the weak formulations are adopted.

11. What do you mean natural coordinates?

12. What are equivalent nodal forces?

13. What are the types of non-linearity in structural analysis?

14. Write down the shape functions for four noded rectangular elements?

15. What are incompatible displacement models?

16. Write the natural co-ordinates for the point P of the triangular element. The point P is the

C.G of the triangle.

17. What are the properties of stiffness matrix?

18. What are the steps involved in finite element modeling?

19. Write the shape function for constant strain triangle by using polynomial function?

20. What are the conditions for a problem to be axisymmetric?

PART-B

1. Derive shape functions and stiffness matrix for a 2D rectangular element.

2. Evaluate the nodal load vector due to self-weight of a four noded rectangular element with

two degrees of freedom (translations) at each node. Use Gauss quadrature method of

numerical integration.

3. Evaluate the shape functions N1, N2,N3 at the interior point P(3.85,4.8) for the triangular

element shown in fig 1.

y 3(4,7)

2(7,3.5)

1(1.5,2)

x

4. Discuss the convergence requirements of interpolation polynomials.

5. The nodal coordinates of a quadrilateral element as given as (0,7), (9,4),(7,9) and (2,8).

Evaluate the integral ʃ (x2 +y

2 -3xy) dA over the area of the element of second order

Gaussian quadrature.

6. Derive shape functions for constant strain triangle element.

7. Integrate the following function using Gaussian integration. Proper Gauss points should be

specified. The x limit is varying from 0 to 2 and y limit is varying from 1 to 3 ʃʃ(xy) dxdy.

8. i) Derive the weights and Gauss points of Gauss one point formula and two point formula.

ii) Derive the shape functions of a four noded quadrilateral element. 1 1

9. i) Evaluate ʃ ʃ (x2

+xy2)dx dy by Gauss numerical integration.

-1 -1

ii) Derive the element strain displacement matrix of a triangle element.

10. Triangular elements are used for the stress analysis of a plate subjected to in plane loads.

The components of displacements parallel to (x, y) axis at the modes 1, 2, 3 are found to be (-

0.001, 0.01), (-0.002, 0.01) and (-0.002, 0.02) cm respectively. If the (x, y) coordinates of the

nodes shown in Fig are in cm, find the components of displacement of the point (30, 25) cm.

y 3(40,40)

1(20,20) 2(40,20)

x

UNIT-III

1. State the conditions to be satisfied in order o use axisymmetric elements?

2. What is meant by error evaluation in FEM?

3. What is meant by an Isoparametric element?

4.Define the Lagrange interpolation polynomials used for higher order elements.

5.Write shape functions for 1 D linear strain element.

6. What is the difference between h and p methods?

7.Brief the application of higher order elements.

8.What is an ill conditioned element?

9. Brief the applications of higher order elements.

10.What are the methods used for numerical integration in finite element method?

11.How the discretization error is evaluated?

12. What is a lumped mass system?

13. What is dynamic condensation?

14. What do you mean by higher order element?

15. What is ILL conditioned element?

16. In an element the geometry is defined using 4 nodes and the displacement is defined

using 8 nodes. What is this element called?

17. When Hermite interpolations functions are used?

18. What are the ways in which 3D problem can be reduced to a 2D approach?

19. What are the types of meshes?

20. Write down the displacement equation for an axisymmetric element?

PART-B

1. Determine the stiffness for the axisymmetric element shown in fig. Take E as 2.1x 105

N/mm2 and Poisson’s ratio as 0.3.

3

1 2

2. Explain the isoparametric elements and its types.

3. Derive the displacement interpolation matrix, strain displacement interpolation matrix B,

and Jacobian operator J for the three node truss element shown in Fig . Also sketch the

interpolation functions.

r =-1 r=0 r=+1

x,u

X1 L/2 L/2

4. State the need for mesh refinement. Discuss the methods of mesh refinement.

5. Derive the shape functions for element shown in fig. Shape functions should be specified

in natural coordinate system.. 7 8 9

η

η

£ 4 555555556

1 2 3

6. Derive the shape functions for ID cubic element. Shape functions should specified in both

natural and global coordinate systems.

7. i)Obtain the shape functions of a nine noded quadrilateral element.

ii) Describe auto and adaptive mesh generation techniques.

8. i) Comment on discretization error with an example.

ii) Discuss p and h methods of refinement and give applications of each method.

9. Determine the shape functions of six noded triangular elements.

10. What is adoptive meshing? Explain any one algorithm for auto meshing.

UNIT-IV

0,0

0,50

50,0

1. List out the meshing techniques?

2. Give two examples of geometric nonlinear problems?

3. List the sources of errors in finite element analysis.

4. List the methods used for evaluation of Eigen values and Eigen vectors.

5.Zero stress gradient is observed along x direction from the following structural model. How

many elements you will consider for your analysis?

σo σo

6. How is geometry nonlinearity taken care in finite element analysis?

7. What do you mean by material non linearity?

8. What is a mass index?

9. Give examples of thermal analysis problems.

10. Give one dimensional heat floe equation.

11. Why higher order elements are necessary?

12. What are the types of non-linearity?

13. What is discretization error?

14. What is weak formulation?

15. How error is evaluated in finite element analysis?

16. What are the types of non-linearity?

17. Form the consistent and lumped mass matrix for a truss element. Length = 3m, Area = 20

x 10-4 m

2 and mass density = 2.5x10

4 kgm

-3

18. When the equilibrium equations are established with respect to the deformed shape, then

the system is analyzed as ---------- nonlinear one.

19. What are isoparametric elements?

20. Give the stiffness matrix for an axisymmetric triangular element.

PART-B

1. Explain the automatic mesh generation technique.

2. Write a detailed note on Matrix solution techniques and Natural coordinate systems.

3. Explain the problems involved in the analysis of material non linearity and explain how a

solution procedure for search problems may be established for structures made of ductile

materials.

4. Explain how the consistent mass matrix for a pinpointed bar element is obtained.

5. Discuss the vector iteration method of Eigen value problems.

6. The cross section of a bar of length 10cm is rectangular in section of width 3cm and depth

1 cm. The bar is subjected to forced convection over its length due to flow of fluid at

temperature of 25OC. The convection coefficient is 5 W/cm

2o C. Compute the thermal load

vector due to convection.

7. Determine Pcritical for a pinned –pinned column of length, L. Consider 3 elements for the

analysis. And also draw mode shape.

8.i) Explain characteristic polynomial technique of Eigen value- Eigen vector evaluation.

ii) Explain any one method of handling geometric non- linearity.

9. Determine the Eigen values and Eigen vectors for the stepped bar shown in fig. Take

E= 2 x 105 N/mm

2 specific weight = 78.5 kN/m

3.

250mm 125mm

10. Using lumped mass approach obtain natural frequencies and shapes of flexural modes of

a fixed end beam of span 900mm.

A= 500mm2, I= 400mm

2, p=7840 kg/m

3 and E=200 kN/mm

2

UNIT-V

1. List out the two advantages of post processing.

2. List out FEM software packages.

3. Write the equation for calculating element mass matrix in terms of shape functions.

4. State the functions of a preprocessor in FEA software package.

5. Name few software packages used for finite element analysis.

6. Specify degrees of freedom for SOLID 45 element analysis.

7. Give one dimensional heat flow equation.

A1=600mm2 A2=300mm2

8. Give examples of thermal analysis problem.

9. What is dynamic condensation?

10. What are the suitable methods for thermal analysis?

11. What is the governing differential equation for a one dimensional heat transfer?

12. What is natural coordinate system?

13. What is lumped mass matrix?

14. How thermal loads are input in finite element analysis?

15. In buckling analysis the Eigen value and Eigen vectors are calculated. Actually what do

they represent?

16. Define Radiation of heat transfer.

17.Brief the applications of higher order element.

18.What is the difference between h and p methods.

19. What is an ill conditioned element.

20.Write discretization errors.

PART-B

1. Explain the generation of node numbers in FE analysis using soft wares.

2. Write a detailed note on mesh plotting.

3.Discuss the modeling procedure using soft wares by taking an example of a plate bending

problems.

4. Write short notes on:

i) One dimensional heat transfer problems in finite element analysis.

ii) Error evaluation in FEA

5.Plate with small centre hole (3mm diameter) is subjected to 50 N tensile load( refer fig).

Thickness of the plate is 6mm and width of the plate is 28mm. Take E= 210 GPa and Poisson

ratio = 0.3. How will you solve this problem using finite element software (ANSYS)?

Determine steps should be provided.

1m

6. How will you solve this problem (refer fig) using finite element software? Detailed steps should be clearly specified. P

L

7.Write the step by step procedure of solving a structural problem by using any finite element

software.

8. Obtain the element matrices for the one dimensional heat conduction equation.

d/dx (KdT/dx)+Q=0 subject to boundary conditions

T/(x=0) = To q/(x=c) =n (TL-T∞)

9. Find the forces developed at the fixed end A due to rise of 60oC.

A= 800mm2

E= 200 kN/mm2

α =10x 10-6

/oC

A 2m B

10. Find the forces in the members due to rise in temperature of 30oC.

A= 400mm2, E= 200kN/mm

2 and α = 70x 10

-6/oC.

C

2m

A B